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House Speaker Paul D. Ryan says the next Congress will pursue a repeal of Obamacare immediately, even before Republicans have a replacement in hand, though that repeal will not pull the rug out from 20 million-plus Americans who gained coverage from President Obama’s signature program.

The road map outlined by Mr. Ryan, Wisconsin Republican, signals that Republicans majorities and President-elect Donald Trump are eager to fulfill their campaign vow to gut the Affordable Care Act in the new year, though any GOP action’s effect will be largely symbolic if many of Obamacare’s provisions remain in place for months — or years — to avoid disruption.

“We want to make sure that we have a good transition period, so that people can get better coverage at a better price,” Mr. Ryan told CBS’ “60 Minutes” in an interview that aired Sunday.

The exact timeline remains unclear.

“I don’t know the answer to that right now,” Mr. Ryan said. “What we know is we have to make good on this promise, we have to bring relief as fast as possible to people who are struggling under Obamacare.”

Mr. Ryan said his long-sought overhaul of Medicare, however, will have to wait until after Congress takes on issues such as tax reform.

His push to turn the popular, government-run insurance program for seniors into a “premium-support” is a political lightning rod that could cause headaches for GOP candidates facing election in 2018, and it didn’t win over Mr. Trump on the campaign trail.

Yet Mr. Trump’s pick to lead the Health and Human Services Department — Republican congressman and House Budget Committee Chairman Tom Price of Georgia — endorsed Mr. Ryan’s vision on Medicare in fiscal blueprints he wrote as chairman of the House Budget Committee.

Under their vision, the current, open-ended entitlement would be replaced with a fixed contribution to help each enrollee buy private insurance. The Medicare program covers about 55 million Americans age 65 and older and younger people with disabilities.

House Minority Leader Nancy Pelosi is already warning the GOP it will pay a political price for fiddling with Medicare, and that her troops will fight attempts to move away from the program’s current model “with all of our strength.”

“With Chairman Price headed to HHS, Speaker Ryan moves closer to realizing his dream and America’s nightmare, of shattering the Medicare guarantee,” she said at her weekly press conference.

Though its Medicare plans are on ice for now, the GOP is charging ahead with efforts to dismantle Mr. Obama’s health legacy.

“No amount of tinkering or taxpayer bailouts is going to make Obamacare work. It needs to be replaced and repealed. Repeal is only the first step — it clears the path for a replacement that costs less and works better than what we have now,” Sen. John Barrasso, Wyoming Republican, said.

While the message is largely the same, the mission will include new players on House committees that will oversee consumer and tax provisions related to health reform.

Rep. Kevin Brady, Texas Republican and chairman of the tax-writing Ways and Means Committee, will now be work alongside ranking member Richard Neal, Massachusetts Democrat, instead of longtime Rep. Sander Levin of Michigan.

Mrs. Pelosi said her caucus is ready to work with Republicans on improving Mr. Obama’s health reforms, but has little desire to start from scratch.

“We’re not going to be party to the dismantling of the Affordable Care Act,” Mrs. Pelosi said.